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Random spatial growth with paralyzing obstacles

J. van den Berg, Y. Peres, V. Sidoravicius, M. E. Vares (2008)

Annales de l'I.H.P. Probabilités et statistiques

We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no ‘inter-green’ competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as a red...

Random walk in random environment with asymptotically zero perturbation

M.V. Menshikov, Andrew R. Wade (2006)

Journal of the European Mathematical Society

We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on ℤ + = { 0 , 1 , 2 , ⋯ } , with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different from the previously studied cases. Our method is based on a martingale technique—the method of Lyapunov functions.

Random Walks in Attractive Potentials: The Case of Critical Drifts

Dmitry Ioffe, Yvan Velenik (2010)

Actes des rencontres du CIRM

We consider random walks in attractive potentials - sub-additive functions of their local times. An application of a drift to such random walks leads to a phase transition: If the drift is small than the walk is still sub-ballistic, whereas the walk is ballistic if the drift is strong enough. The set of sub-critical drifts is convex with non-empty interior and can be described in terms of Lyapunov exponents (Sznitman, Zerner ). Recently it was shown that super-critical drifts lead to a limiting...

Regularity of the effective diffusivity of random diffusion with respect to anisotropy coefficient

M. Cudna, T. Komorowski (2008)

Studia Mathematica

We show that the effective diffusivity of a random diffusion with a drift is a continuous function of the drift coefficient. In fact, in the case of a homogeneous and isotropic random environment the function is C ∞ smooth outside the origin. We provide a one-dimensional example which shows that the diffusivity coefficient need not be differentiable at 0.

Reinforced walk on graphs and neural networks

Józef Myjak, Ryszard Rudnicki (2008)

Studia Mathematica

A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.

Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment

Christophe Sabot, Laurent Tournier (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ ℓ →n +∞ for some ℓ) with positive probability. In dimension 2, this result is strenghened...

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